 1585 R. CampoamorStursberg
 Deformations of Lagrangian systems preserving a fixed subalgebra of Noether symmetries
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Aug 25, 15

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Abstract. Systems of secondorder ordinary differential equations admitting
a Lagrangian formulation are deformed requiring that the extended
Lagrangian preserves a fixed subalgebra of Noether symmetries of
the original system. For the case of the simple Lie algebra
$rak{sl}(2,\mathbb{R})$, this provides nonlinear systems with
two independent constants of the motion quadratic in the
velocities. In the case of scalar differential equations, it is
shown that equations of Pinneytype arise as the most general
deformation of the timedependent harmonic oscillator preserving
a $rak{sl}(2,\mathbb{R})$subalgebra. The procedure is
generalized naturally to two dimensions. In particular, it is
shown that any deformation of the timedependent harmonic
oscillator in two dimensions that preserves a
$rak{sl}(2,\mathbb{R})$ subalgebra of Noether symmetries is
equivalent to a generalized ErmakovRayReid system that satisfies
the Helmholtz conditions of the Inverse Problem of Lagrangian
Mechanics. Application of the procedure to other types of
Lagrangians is illustrated.
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